Words that Ever Play Well

Review

In Part I we introduced Negative Polarity Items, a special class of words that just don’t seem to ever play right. We tried to tackle the question “in what contexts can we use NPIs?”.

Ultimately we came up with (what I would call) a pretty compelling answer; downward entailment is when something you know about \(X\) tells you something about a subset of \(X\).

1. No dog can fly. \(\vDash\) No talking dog can fly.
2. A talking dog can fly. \(\vDash\) A dog can fly.

Because “talking dogs” \(\subseteq\) “dogs”, we call (1) a downward entailment (or DE) and (2) an upward entailment (or UE). And the magic is that DEs licence our troublesome NPIs! (Ladusaw 1996)

In the end, however, we were left with a problematic sentence:

This sentence is neither upward nor downward entailing. In this post, we’re going to do something about that!

If you’re reading this, I’m assuming Part I already got your attention. So, if you’ll permit me, I’m going to get a little more technical here. Time to put on our big-kid-amateur-semanticist pants!

The Big Picture

Before we get started, let’s take a deep breath and ask ourselves, “why does this matter?” It’s fun, sure, but does it have a real impact on our ideas of language?

Well-Formedness

To answer your question:

3. Colorless green ideas sleep furiously.

Noam Chompsky said that, and I think he knows a thing or two!¹ It was to demonstrate (among other things) there exists well-formed sentences that are meaningless. So what do I mean by well-formed?

4. #Furiously sleep ideas green colorless.

Your brain was able to “read” (3). Were I to ask you “what sleeps?” you could respond “colorless green ideas… whatever that means.” Example (4), however, hardly deserves the title “sentence”. It is not grammatically well-formed. Let’s look at some other examples:

NPI Errors

What about “I found someone” vs “#I found anyone”? Is that a failure of meaning, like (3), or of grammar, like (4)? Qualitatively, it feels closer to a grammar problem than a semantic problem — it has that “it hurts my ears” quality to it.⁵

And that is my (somewhat abstract) point — we have been talking about NPI errors as semantic in nature, having to do with what a sentence means, rather than the structure of the sentence itself. Well-formedness, however, is thought of as being essentially syntactic. NPI errors bring into question whether semantics play a roll in well-formedness.

Situations Not Covered by DE

OK, back to the task at hand!

We’ve already found one example where DE seemed to fail us. We labeled it as non-monotonic, meaning neither UE nor DE. If we’re going to fix DE, we should understand just exactly when it fails.

Questions

Questions don’t entail much of anything. If I ask “were is the milk”, at best you could infer that I don’t know where the milk is (and that I would sure like some). It tells you nothing about the more specific blue milk; it tells you nothing about less specific fluids. Questions are neither upward or downward entailing.

It actually seems that NPIs are most used in questions (across all languages), so this is a real problem for us!

Imperatives

Subtly, this isn’t DE — no one asked you to “pick any blue painting”. You may, but that would be a different sentence.

Others

In Giannakidou (2011, sec. 3.2.3), a number of other examples are laid out, including:

Most are non-obvious why they don’t fall under DE, and some don’t have English equivalents. The take away is, DE doesn’t always seem to work.

A Solution

We started things off with the interface between semantics and grammar. Then we examined some sentences which are not downward entailing, but which do allow NPIs. Here’s what I want to do now:

1. Define a funky little term
2. Show anything DE can explain, our new term covers
3. Demonstrate it solves our problems

And the funky little term of the day is (non)veridicality!

(Non)veridacity

Veridicality is related to the truth or existence of something. Before we define this, let’s go strait to some examples:

Ok, that’s not so bad. So is the sentence veridical? Or just a word?

“Found”, “looking”, and “there is no” can all be called operators. Here, they operate on the existence of \(p\), the meaning of life. If I “find” some \(p\), then I know \(p\) exists (written \(O_\text{find}p \to p\) ).⁷

For practice, let’s see what these sentences say (or don’t say) about \(p\):

Intuitively, if you want to know if something is veridical, ask yourself “does this imply that something exists?” But if we’re going to use veridicality to solve our NPI woes, we’re going to need an accurate definition.

Veridical

An operator \(O\) is veridical if \(Op \to p\).

If we test the NPI any, we can get an intuition about how this will work:

The claim is that nonveridacity licences NPIs.

DE \(\subseteq\) Nonveridacity

So how are nonveridacity and downward entailing related? They seem very different. Zwarts (2003) actually provides a proof that all DE operators are also nonveridical. Walking through it isn’t too difficult.

We’ll look at operators that take two arguments — things like “and” (\(p \wedge q\)), “or” (\(p \vee q\)), “if/then” (\(p \to q\)), “with”, “without”, etc. We call these connectives.

Veridical Connective

A connective \(C\) is said to be veridical (with respect to \(p\)) if \(pCq \to p\). That is, if one knows \(pCq\), then one can infer \(p\).
Example:
\(p \wedge q \to p\)
If you know “\(p\) and \(q\)”, then you must know \(p\)!

So how, then, do a DE connectives look like?

Downward Connective

Let \(C\) be downward entailing with respect to \(p\). Knowing \(r \to p\) (whenever \(r\) is true, \(p\) is true) is enough to know \(pCq \to rCq\).
Example:
“I will kill you” \(\to\) “You will die”
“If you sleep with anyone, I will kill you” \(\to\) “If you sleep with anyone, you will die”
“If \(p\) then \(q\)” is DE with respect to \(q\).

This is really the same definition we used in Part I, but now we are saying \(r \to p\), rather than \(R \subseteq P\). Showing that DE \(\subseteq\) Nonveridacity is now very strait-forward. We can prove that no connector \(C\) can be both DE and veridical.

1. Let \(C\) be veridical and DE.
2. Assume \(pCq\).
3. Note that, because \(a \wedge b \to a\), \(p \wedge \neg p \to p\).
4. Because \(C\) is DE, we know \((p\wedge \neg p)Cq\).
5. Because \(C\) is veridical, we know \((p\wedge\neg p)\).

This is obviously a contradiction, therefore \(C\) was impossible!⁸ That means that any connector that is DE must be nonveridical.

Nonveridacity Solves Our Problems

Last step: show that nonveridacity actually solves our problems! Everything in Part I is covered, but what about the new problems?

Questions

A question, inherently, does not imply truth — if you already knew something was true, why are you asking?

Commands

The same goes for commands. “Name one good Matrix Sequel” — see, you can’t! That sentence didn’t say such a thing exists (almost the opposite).

“#I found anyone” doesn’t work, as it claims that anyone exists. But “find anyone” is still a command, even if there is no one to find.

Quantifiers

Think about what each quantifier really means:

DE doesn’t cover this discrepancy, but nonveridicality seems to have our back!

The Jury Isn’t Out

DE does a pretty good job predicting when NPIs can be used. Nonveridicality does an even better job. But what about our original problem child?

I… Uh… Oh no…

All Kinda of Polarity!

The discovery of DE excited everyone — finally we could put all these NPIs in a nice clean box, and put a bow on it. The reality, however, is more complicated. Especially when we started examining more and more languages.

Maybe most importantly, there are multiple kinds of polarity items!

Each type has related but different rules that govern their usages. Especially looking at other languages casts light on the differences that come up; each language has troves of examples. Here’s one, for instance:

Our problem child behaves as expected in Greek! Maybe… maybe that’s going to have to be good enough for today.

And here is where I leave you. Maybe a simple solution is out there. Then again, it could just be that language is an amazingly complex beast.

Parting Notes

Thanks for following me through this little journey. It’s been a lot of fun for me to learn about polarity items, and various ways we can look at them. Most of what I’ve presented here is based on work by Anastasia Giannakidou — if you want more information, I’d suggest starting with her paper in the references. (Giannakidou 2011)

It’s a shame we only worked through English examples. If I get enough requests, I might make a third post demonstrating some NPIs in other languages — whatever you speak, I’m sure they’re there.

References